arXiv:1511.07048v2 [math.AG] 28 Jul 2017 e,b sn h M.Trecssa olw ih cu,acc occur, might follows as cases Three of MMP. the using by nef, nt eaecvr If ´etale cover. finite a have n n[P7 o h l arcs.Teeoe opoe(1.1. prove to Therefore, case. pair klt the for [CP17] in and 3srae nodrt ra hscs,w eyo w aning main two on rely we case, this images treat direct to the order In surface. K3 eei bradlet and fiber generic aaaai Kw2] oepeiey ehv h following the have we precisely, More [Kaw82a]. in Kawamata nti oe epoeta h o-eso fItk conject Iitaka of log-version the that dim prove we note, this In fiber. emtyi the is geometry where asoffe g´eom´etrie en k¨ahl´erienne”Hausdorff (ANR-GRACK). Let Keywords: Classification Subject Mathematics 2010 1.1 hswr sprilyspotdb h gneNtoaed l de Nationale Agence the by supported partially is work This Y . If If ewl etepantemi tp ftepof ic dim Since proof. the of steps main the explain next will We p Y . surfaces. nti hr oew rv h Iitaka the prove we note short this In oar ieso fagbacfie pcsover spaces fiber algebraic of dimension Kodaira Theorem. κ κ : κ 6 ( ( X ( Y Y X ;ti eeaie eutotie yC ikri [Bir09, in Birkar C. by obtained result a generalizes this 2; iaacnetr,pstvt fdrc images direct of positivity conjecture, Iitaka ) ,b h lsicto hoyw nwthat know we theory classification the by 0, = ) → steKdiadmninof dimension Kodaira the is ) > Y ,teieult 111 sqikyvrfidb sn [Kaw82 using by verified quickly is (1.1.1) inequality the 1, eafirto ewe w rjciemnfls eta pr central A manifolds. projective two between fibration a be iaaconjecture Iitaka Let p ⋆ ( mK ∆ p Y : ea be X satrs 111 spoe n[H1 o h bouecs ( case absolute the for [CH11] in proved is (1.1.1) torus, a is X/Y → Q + κ Y eetv l iio on divisor klt -effective m ( K ttn that stating , eafirto ewe w rjciemnfls Let manifolds. projective two between fibration a be )adtegoer fobfl aaiYusrae.Differen surfaces. Calabi-Yau orbifold of geometry the and ∆) X 43,3J5 40,14J40 14D06, 32J25, 14E30, κ ∆) + ( X .Introduction 1. ) uynCao Junyan X surfaces > > Abstract and κ C κ ( ( nm Y K κ + ) F ( ojcuefragbacfie pcsover spaces fiber algebraic for conjecture X/Y ∆ + κ ( X/Y X F steKdiadmnino h generic the of dimension Kodaira the is ) Set . + ) ehrh rn Cnegned Gromov- de “Convergence grant Recherche a r od re rvddta h base the that provided true, holds ure Y ) ti nuht suethat assume to enough is it 1), (1.0.1) ) κ satrso 3srae modulo surface, K3 a or torus a is ( ∆ Y Y statement. eins aeytepstvt of positivity the namely redients, rigt h oar dimension Kodaira the to ording F ) . 6 =∆ := ,w a suethat assume can we 2, h .]adarsl fY. of result a and 1.4] Thm | F If . a]. dim be nbirational in oblem Y 6 .. 0) = ∆ i.e., 2 hnwe then , F (1.1.1) ethe be Y K Y sa is is t aspects of the first topic were extensively studied in [Gri70, Fuj78, Kaw82a, Kaw82b, Kol85, Vie95, Ber09, BP10, PT14, BC15, Fuj16, HPS16, KP17], among many other articles. In our set- up, this implies that det p⋆(mKX/Y + m∆) is pseudo-effective (by using [PT14]). As Y is a K3 surface, the numerical dimension of det p⋆(mKX/Y + m∆) coincides with its Iitaka dimension. If the numerical dimension nd(det p⋆(mKX/Y + m∆)) > 1, we achieve our goals by standard arguments. If nd(det p⋆(mKX/Y + m∆)) = 0, we can show that there exists a finite set of exceptional curves [Ci] on Y such that p⋆(mKX/Y + m∆) is hermitian flat on Y \ (∪Ci), by using the results in [PT14, CP17, HPS16]. At this point we use the second ingredient, namely the uniformization theorem for the compact K¨ahler orbifolds with trivial first Chern class, cf. [Cam04a]. We thus infer that the fundamental group of Y \ (∪Ci) is almost-abelien. Therefore 0 we can construct sufficient elements in H (X,mKX/Y + m∆) by using parallel transport, and (1.1.1) is proved. Acknowledgements: We would like to thank J.-A. Chen who brought my attention to this problem and valuable suggestions on the article. The author would also like to thank M. P˘aun for extremely enlightening discussions on the topic and several important suggestions on the first draft of the article. We are also indebted with H.-Y. Lin for pointing out to us the reference [Cam04a], which plays a key role in this article. Last but not least, we would like to thank the anonymous referees for their valuable suggestions.

2. Preparation

In this section, we will recall the uniformization theorem for the compact K¨ahler orbifolds with trivial first Chern class [Cam04a] (cf. also [CC14, GKP15]) as well as the results concerning singular metrics on vector bundles and the positivity of direct images, cf. [BP08, PT14, Rau15, Pau16, HPS16] for more details. First of all, we recall a few basic definitions concerning compact K¨ahler orbifolds by following [Cam04a]. 2.1. Definition.[Cam04a, Definition 3.1, Definition 5.1] (i) A compact K¨ahler orbifold is a compact K¨ahler normal variety with only quotient singular- ities, i.e., for every point a ∈ X, we can find a neighbourhood U of a, and a biholomorphism ψ : U → U/G where U is an open set in Cn and G ⊂ GL(n, C) is a finite subgroup acting on U withe ψ(a) = 0. Fore every g ∈ G, the set of the fixed points of g is of codimension at leaste 2. (ii) Let X⋆ be the smooth locus of a compact K¨ahler orbifold X. We say that X is simply connected in the sense of orbifolds if X⋆ is simply connected. (iii) A holomorphic morphism between two K¨ahler orbifolds r : X′ → X is said to be an orbifold cover, if it satisfies the following two conditions: • The restriction of r to r−1(X⋆) is an ´etale cover. • For every a ∈ X with its neighbourhood U/G (cf. (i)), each component of r−1(U/G) ′ ′ is of the form U/G for some subgroup G of G, and the restricted morphism r| e ′ is e U/Ge nothing but thee natural quotient morphism U/G′ → U/G. (iv) A m-dimensional compact K¨ahler orbifold X is callede Calabi-Yaue (resp. Hyperk¨ahler), if it is simply connected in the sense of orbifold (cf. (ii)) and it admits a Ricci-flat K¨ahler metric

2 such that the holonomy (when restricted to X⋆) is SU(m) (resp. Sp(m/2)).

We state here the uniformization theorem for the compact K¨ahler orbifolds with trivial first Chern class, established in [Cam04a]. The statement parallels the classical case of smooth K¨ahler manifolds with trivial first Chern class.

2.2. Theorem.[Cam04a, Thm 6.4] Let X be a compact K¨ahler orbifold with c1(X) = 0. Then X admits a finite orbifold cover X = C × S × T , where C (resp. S) is a finite product of Calabi-Yau K¨ahler orbifold (resp. Hyperk¨ahler) and T is a complex torus.

Let Y be a K3 surface and let ∪Ci be a set of exceptional curves on Y . By Grauert’s criterion [BHPV, III, Thm 2.1], there is a contraction morphism τ : Y → Ycan which contracts all Ci to some points pi in a normal space Ycan. As KY is trivial, we know that Ycan is in fact a compact K¨ahler orbifold ([KM98, Def 4.4, Rk 4.21], [Cam04a, Example 3.2]) with c1(Ycan) = 0.

As a corollary of Theorem2.2, we have.

2.3. Proposition.[Cam04a, Cor 6.7] Let Y be K3 surface and let ∪Ci be some exceptional curves on Y . Then π1(Y \ (∪Ci)) is almost-abelien.

Moreover, let τ : Y → Ycan be the morphism which contracts the exceptional curves Ci to some points pi ∈ Ycan. If π1(Y \ (∪Ci)) is not finite, there exists a finite orbifold cover from a complex torus T to Ycan:

σ : T → Ycan. −1 In particular, σ is a non-ramified cover over Ycan \(∪pi) and σ (pi) is of codimension 2 for every i.

The following non-vanishing property for pseudo-effective line bundles on K3 surfaces is an immediate consequence of the abundance theorem (which holds true in dimension two).

2.4. Proposition. Let Y be a K3 surface (in the smooth sense) and let L be a pseudo-effective line bundle on Y . Then L is Q-effective.

Proof. Since L is pseudo-effective, by using Zariski decomposition for surface [Fuj79, Thm 1.12], we know that p

L ≡Q ai[Ci]+ M, Xi=1 + where ai ∈ Q , Ci are negative intersection curves, M is nef and M · Ci = 0 for every i. Since Y is K3, all nef line bundles on Y are effective (cf. [BHPV, VIII, Prop 3.7]). Therefore L is Q-effective.

2.5. Remark. It is well-known that a nef line bundle M on a K3 surface is semiample. If its numerical dimension nd(M) is equal to one, then it induces an elliptic fibration over P1.

In the second part of this section we will recall a few definitions and results about the singular metrics on vector bundles and the positivity of direct images. We refer to [BP08, Rau15, PT14, Pau16, HPS16] for more details.

3 2.6. Definition. Let E → X be a holomorphic vector bundle on a manifold X (which is not necessary compact). Locally, a singular hermitian metric hE on E is a measurable map from X to the space of non-negative Hermitian forms on the fibers. We say that (E, hE ) is negatively curved, if 0 < det hE < +∞ almost everywhere and

x → ln |u|hE (x) x ∈ X is a psh function, for any choice of a holomorphic local section u of E. ⋆ ⋆ We say that the pair (E, hE ) is positively curved, if the dual (E , hE) is negatively curved.

We note it by iΘhE (E)  0.

When hE is smooth, ”positively curved” is nothing but the classical Griffiths semi-positivity. The following result proved in [PT14] plays an important role in this article. 2.7. Theorem.[PT14, Thm 5.1.2] Let p : X → Y be a fibration between two projective manifolds and let L be a line bundle on X with a possibly singular metric hL such that iΘhL (L) > 0. Let 1 m m ∈ N such that the multiplier ideal sheaf I(hL |Xy ) is trivial over a generic fiber Xy, namely 2 |e | m < +∞, where e is a basis of L. Xy L hL L R Let Y1 be the locally free locus of p⋆(mKX/Y + L). Then the vector bundle p⋆(mKX/Y + L) over Y1 admits a possibly singular hermitian metric h such that iΘh(p⋆(mKX/Y + L))  0 on Y1. Moreover, h induces a possibly singular metric det h on the line bundle det p⋆(mKX/Y + L) over Y such that

iΘdet h(det p⋆(mKX/Y + L)) > 0 on Y in the sense of current.

2.8. Remark. Let us recall briefly the construction of the metric h: Let hB be the m-relative 1 Bergman kernel metric on KX/Y + m L constructed in [BP10, A.2]. Set L1 := (m − 1)KX/Y + L 1 and hL1 := (m − 1)hB + m hL. Thanks to [BP10, A.2], we know that > iΘhL1 (L1) 0 on X (2.8.1) in the sense of current. Now hL1 induces a Hodge type metric h on π⋆(mKX/Y + L) on the 0 smooth locus Y0 of π as follows: let Xy be a smooth fiber and let f ∈ H (Xy, mKX/Y + L). As mKX/Y + L = KX/Y + L1, the norm

kfk2 := |f|2 h hL1 ZXy is well defined. Since hL is not necessarily smooth, h is a possibly singular hermitian metric on (π⋆(mKX/Y + L),Y0). Thanks to [Ber09, BP08], we can prove that (p⋆(mKX/Y + L), h) is positively curved on Y0. By studying the comportment of h near Y1 \ Y0, [PT14] proved finally that h can be extended as a possibly singular hermitian metric on Y1 with positive curvature in the sense of Definition2.6.

The following proposition comes from the standard extension theorem. 2.9. Proposition. In the setting of Theorem2.7, we suppose moreover that there exists a fibration q : Y → Z to some projective manifold Z. Let H be a pseudo-effective line bundle on Y with a possible singular metric hH such that iΘhH (H) > 0 in the sense of current. Let AZ be an ample line bundle on Z. Then for c ∈ N large enough (depending only on AZ and Z), the following extension property holds:

4 Let z ∈ Z be a generic point and let Xz (resp. Yz) be the fiber of p ◦ q (resp. q) over z. Let 0 e ∈ OZ,z(c · AZ ) and let s ∈ H (Yz, KY ⊗ H ⊗ p⋆(mKX/Y + L)) such that

2 |s|hH ,h < +∞, (2.9.1) ZYz 1 where h is the metric on p⋆(mKX/Y + L) in Theorem2.7 . Then there exists a section 0 ⋆ S ∈ H (Y, KY ⊗ H ⊗ p⋆(mKX/Y + L) ⊗ q (cAZ )) ⋆ such that S|Yz = s ⊗ q e.

Proof. Let (L1, h1) be the line bundle constructed in Remark 2.8. Then s induces a section 0 u ∈ H (Xz, KX + H + L1) and (2.9.1) implies that

2 |s|hH ,h1 < +∞. ZXz

For c ∈ N large enough (depending only on Z and AZ ), by the standard Ohsawa-Takegoshi extension theorem (cf. for example [Dem12, Chapter 13]), we can find a 0 ⋆ U ∈ H (X, KX + H + L1 + (p ◦ q) cAZ ) ⋆ such that U|Xz = u ⊗ (p ◦ q) e. Then U induces a section 0 ⋆ S ∈ H (Y, KY ⊗ H ⊗ p⋆(mKX/Y + L) ⊗ q (cAZ )) ⋆ such that S|Yz = s ⊗ q e and the proposition is proved. As another direct consequence of the Ohsawa-Takegoshi extension, the following proposition is will be important for us. 2.10. Proposition.[BP10, A.2] In the setting of Theorem2.7, let U be a small Stein open subset of X and let V ⋐ U be some open set of compact support in U. Let e be a basis of mKX/Y + L over U. Then there exists a uniform constant C(U, V, e) depending only on U, V, e such that for 0 every t ∈ π(V ) and every s ∈ H (Xt, mKX/Y + L), we have s k k 0 −1 6 C(U, V, e) · ksk . e C (V ∩π (t)) h

Proof. As explained in Remark 2.8, the line bundle L1 := (m − 1)KX/Y + L can be equipped with a possibly singular metric hL1 such that > iΘhL1 (L1) 0 on X. Since U is a small open set, we can find a Stein open set B ⊂ Y such that U ⊂ p−1(B). As mKX/Y +L = KX/Y +L1, by applying the Ohsawa-Takegoshi extension theorem to the fibration −1 0 −1 p (B) → B, we can find a s ∈ H (p (B), KX + L1) such that

e |s|2 6 C |s|2 = C · ksk2 (2.10.1) −1 hL1 hL1 h Zp (B) ZXt and e ⋆ s|Xt = s ∧ p (eB), (2.10.2) where eB is a basis of KY over B. e 1 ⋆ 2 As q KZ is a trivial bundle on Yz, modulo this trivial bundle, |s|hH ,h can be seen as a volume form on Yz. Therefore the integral (2.9.1) is well-defined.

5 ⋆ On the open set U, s can be written as s = w · e ∧ p (eB) for some holomorphic function w on U. Note that V ⋐ p−1(B), (2.10.1) implies thus that e e e e kwkC0(V ) 6 C(U, V, e) · kskh s for some constant C(U, V, e) dependinge only on U, V and e. Thanks to (2.10.2), we have w|Xt = e . Therefore s e k k 0 6 C(U, V, e) · ksk . (2.10.3) e C (V ∩Xt) h The proposition is proved.

The last result of this section concerns the regularity of the metric h.

2.11. Proposition.[CP17, Cor 2.8] Let E → X be a holomorphic vector bundle on a manifold X (which is not necessary compact). Let hE be a possibly singular hermitian metric on E such that (E, hE ) is positively curved. Let U be a topological open set of X. If

iΘdet hE (det E) ≡ 0 on U, then hE is a smooth metric on E|U , and (E|U , hE) is hermitian flat.

3. Proof of the main theorem

We now prove the main theorem of the article.

3.1. Theorem.[=Theorem1.1] Let p : X → Y be a fibration between two projective manifolds. Let F be the generic fiber and let ∆ be a Q-effective klt divisor on X. Set ∆F := ∆|F . If dim Y 6 2, then

κ(KX + ∆) > κ(KF +∆F )+ κ(Y ). (3.1.1) Proof. Since Y is of dimension 2, we can consider its minimal model and assume that Y is a smooth projective surface with nef canonical bundle. We show next that it will be enough to treat the case where Y is a K3 surface.

Indeed, if κ(Y ) > 1, as the klt version of Cn,1 is known (cf. [Kaw82a, CP17]), we have thus (3.1.1). We refer to Proposition4.2 in the appendix for a detailed proof. If κ(Y ) = 0, by using 1,1 the classification of minimal surface [BHPV, Thm 1.1], we have c1(Y ) = 0 ∈ HQ (Y ). After a finite ´etale cover2, the base Y is either a torus or a K3 surface. If Y is a torus, [CP17, Thm 1.1] implies (3.1.1). We assume in this way for the rest of our proof that Y is a K3 surface.

Let m ∈ N be sufficiently divisible and let Y1 be the locally free locus of the direct image sheaf p⋆(mKX/Y + m∆). By using Theorem2.7, there exists a possibly singular hermitian metric h on (p⋆(mKX/Y + m∆),Y1) such that

iΘh(p⋆(mKX/Y + m∆))  0 on Y1, (3.1.2) and h induces a hermitian metric det h on (det p⋆(mKX/Y + m∆),Y ) such that

iΘdet h(det p⋆(mKX/Y + m∆)) > 0 on Y in the sense of current. In particular, the bundle det p⋆(mKX/Y + m∆) is pseudo-effective.

2We remark that (3.1.1) is invariant under this operation.

6 By Proposition2.4, we have a Zariski decomposition s det p⋆(mKX/Y + m∆) ≡Q ai[Ci]+ Lm, (3.1.3) Xi=1 + where ai ∈ Q , [Ci] are negative intersection curves, Lm is nef and Lm · Ci = 0 for every i. Let nd(Lm) be the numerical dimension of Lm. We will distinguish next among three cases, according to the numerical dimension of Lm.

Case 1: The numerical dimension of Lm equals two

Then we infer that the bundle det p⋆(mKX/Y + m∆) is big on Y , and (3.1.1) is thus proved by using [Cam04b] or [CP17, Thm 5.1].

Case 2: The numerical dimension of Lm equals one 1 Thanks to Remark 2.5, Lm is semiample. Then Lm induces a fibration π : Y → P . As Lm · [Ci] = 0 for every i, we have

Lm · det p⋆(mKX/Y + m∆)=0. (3.1.4) By using [Vie83, Lemma 7.3], we can find a birational morphism Y ′ → Y from a projective ′ ′ ′ manifold Y , and a desingularisation X of Y ×Y X satisfying: π X′ −−−−→X X

p′ p     Yy′ −−−−→ Yy πY

π   Py1 ′ ′ each divisor W ⊂ X such that codimY ′ p (W ) > 2 is πX -contractible. Since ∆ is klt, we can find ′ ′ ′ a klt Q-effective divisor ∆ on X and some effective πX -exceptional divisor D such that

⋆ ′ ′ ′ πX (KX +∆)+ D = KX +∆ . (3.1.5)

Claim. The bundle ′ ′ ⋆ det p⋆(mKX′/Y ′ + m∆ ) − c(πY ◦ π) OP1 (1) is pseudo-effective on Y ′ for some constant c> 0. We will verify this claim later; for now we finish the proof of the theorem. By using [CP17, Thm ′ ′ 3.4], the claim implies the existence of a divisor E ⊂ X such that codimY ′ p (E) > 2 and ′ ′ ⋆ D := KX′/Y ′ +∆ + E − ǫ(p ◦ πY ◦ π) OP1 (1) (3.1.6) is Q-pseudo-effective on X′ for some ǫ> 0.

Let m1 ≫ m2 ≫ 1. Thanks to (3.1.6), we have ′ (m1 + m2)(KX′/Y ′ +∆ + E) (3.1.7)

′ m2 ′ ⋆ = m1(KX′/Y ′ +∆ + D + E)+ εm2(p ◦ πY ◦ π) OP1 (1). m1

7 2 ′ ′ ′ m As m1 ≫ m2, we can apply Theorem2.7 to m1(KX /Y +∆ + m1 D + E). In particular, we can find a possibly singular metric hm1 on

′ ′ m2 V1 := p⋆(m1(KX′/Y ′ +∆ + D + E)) m1 > ′ such that iΘhm1 (V1)  0. Set T := iΘdet hm1 (det V1). Then T 0 in the sense of current. Let Yt be a generic fiber of πY ◦ π.

If T | ′ is not identically 0, as Y ′ is of dimension 1, T | ′ is strictly positive at a generic point Yt t Yt ′ ′ ′ ′ of Yt . Together this with (3.1.7), det p⋆((m1 + m2)(KX′/Y ′ +∆ + E)) is big on Y . By applying [CP17], we get ′ ′ κ(X , KX′ +∆ + E) > κ(F, KF +∆F ) + 2. (3.1.8) ′ As E and D are πX -contractible, (3.1.5) and (3.1.8) imply (3.1.1).

′ ′ ′ If T | ≡ 0, thanks to Proposition 2.11, (V | , h 1 ) is hermitian flat on Y . In particular, Yt 1 Yt m t ′ 0 ′ ′ h 1 | is a smooth metric. Note that H (Y , K ) is of dimension 1. It defines a canonical metric m Yt Y ′ ′ ′ ′ hY on KY and the restriction of hY on Yt is smooth. As a consequence, we have

2 0 ′ ′ ′ |s| 1 ′ < +∞ for every s ∈ H (Yt , KY ⊗ (m1 − 1)KY ⊗V1). Z ′ (m −1)hY ,hm1 Yt Combining this with Proposition2.9 3, we get

0 ′ ⋆ h (Y , KY ′ ⊗ (m1 − 1)KY ′ ⊗V1 ⊗ εm2(πY ◦ π) OP1 (1))

0 ′ 0 ′ ′ m2 > h (Y , K ′ ⊗ (m − 1)K ′ ⊗V )= h (X ,m (K ′ +∆ + D + E)). t Y 1 Y 1 t 1 Xt m1 Together with (3.1.7), we obtain

0 ′ ′ ⋆ ′ 0 ′ ′ m2 h (X ,m ·(p ) K ′ +(m +m )(K ′ ′ +∆ +E)) > h (X ,m (K ′ +∆ + D+E)). (3.1.9) 1 Y 1 2 X /Y t 1 Xt m1

′ ′ Finaly, by applying [Kaw82a, CP17] to Xt → Yt , we have

′ ′ m2 κ(X , K ′ +∆ + D + E) > κ(F, K +∆ ). t Xt F F m1

Together with (3.1.9) and the fact that KY ′ is Q-effective, we obtain ′ ′ κ(X , KX′ +∆ + E) > κ(F, KF +∆F ). (3.1.10) ′ As E and D are πX -contractible, (3.1.5) and (3.1.10) imply (3.1.1).

Case 3: The numerical dimension of Lm equals zero

Then Lm is trivial (as it is semiample) and we have s det p⋆(mKX/Y ) ≡Q ai[Ci] (3.1.11) Xi=1

3 We take H =(m1 − 1)KY ′ and hH =(m1 − 1)hY ′ .

8 where [Ci] are negative curves. As iΘdet h(det p⋆(mKX/Y +m∆)) is a positive current in the same s class of ai[Ci], we get iP=1 s iΘdet h(det p⋆(mKX/Y + m∆)) = ai[Ci] on Y Xi=1 in the sense of current. In particular, we have

iΘdet h(det p⋆(mKX/Y + m∆)) ≡ 0 on Y \ (∪Ci).

By using Proposition2.11, (p⋆(mKX/Y + m∆), h) is hermitian flat on Y1 \ (∪Ci).

Let : τ : Y → Ycan be the morphism which contracts the negative curves ∪Ci. There are two possible cases: π1(Y \ (∪Ci)) is finite or infinite. We will analyze each possibility.

3.0.1 The fundamental group π1 Y \ (∪Ci) is finite. As codimY (Y \ Y1) > 2, we know that π1(Y1 \ (∪Ci)) = π1(Y \ (∪Ci)) is finite. Let
r be the number of elements of the finite group π1(Y1 \(∪Ci)). Fix a generic point y ∈ Y1 \(∪Ci). As the direct image vector bundle (p⋆(mKX/Y + m∆), h) is hermitian flat on Y1 \ (∪Ci), the parallel transport induces a representation 0 ρ : π1(Y1 \ (∪Ci)) → Aut(H (Xy, mKX/Y + m∆)). (3.1.12) 0 Let f ∈ H (Xy, mKX/Y + m∆) be an element with unit norm. Although the parallel transport of f cannot induce a global section over Y1 \ (∪Ci), the corresponding parallel transport of

0 ρ(a)(f) ∈ H (Xy, mr(KX/Y + ∆)) a∈π1(Y1\(∪Ci))

0 −1 induces a section f ∈ H (p (Y1 \ (∪Ci)), mr(KX/Y + ∆)). e We now prove that f can be extended to the total space X. Let U be an arbitrary small Stein open subset of X and V ⋐ U be some arbitrary open set with compact support in U. Let e be e ⊗r a basis of mKX/Y + m∆ on U. We have f = l · e for some holomorphic function

0 −1 e e −1 l ∈ H (V ∩ p (Y1 \ (∪Ci)), OV ∩p (Y1\(∪Ci))).

e r 0 By construction, on every fiber Xt, f = i=1 fi for some fi ∈ H (Xt, mKX/Y + m∆) with unit f 0Q i 0 norm. Thanks to Proposition2.10, theeC -norm k e kC (V ∩Xt) is bounded by a constant C(U, V, e) independent of t. Therefore r fi r 0 0 6 klkC (V ∩Xt) = k kC (V ∩Xt) C(U, V, e) . Y e e i=1 −1 In particular, |l| is bounded on V ∩ p (Y1 \ (∪Ci)) and f can be thus extended as a holomorphic section on V . Sincee V is an arbitrary small open set in Xe , f can be extended to the total space X. e

0 In conclusion, for any element f ∈ H (Xy, mKX/Y + m∆), we can find a

0 f ∈ H (X, mr(KX/Y + ∆)) e

9 such that f|Xy = a∈π1(Y1\(∪Ci)) ρ(a)(f). In particular, we have e Q div(f|Xy )= div(ρ(a)(f)). X e a∈π1(Y1\(∪Ci)) Therefore, κ(KX + ∆) > 1 if κ(KF +∆F ) > 1. In other words, we have

κ(KX + ∆) > min{1, κ(KF +∆F )}. Together with a standard argument (cf. Proposition4.1 in the appendix), we get

κ(KX + ∆) > κ(KF +∆F ) and the first subcase is completely proved.

3.0.2 The fundamental group π1 Y \ (∪Ci) is not finite. As a consequence of Proposition 2.3, there exists a orbifold cover from a complex
torus T to Ycan:

τY : T → Ycan. ′ Let X be a desingularisation of X ×Ycan T . We have thus a commutative diagram τ X′ −−−−→X X

p′ p     Ty −−−−→ Yycan τY −1 Set T1 := τY (τ(Y1 \ (∪Ci))), where τ : Y → Ycan is the contraction morphism. Thanks to Proposition2.3, τY is a non-ramified cover on T1 and

codimT (T \ T1) > 2. As ∆ is klt, we can find a klt Q-effective divisor ∆′ on X′ and some Q-divisor D′ supported in ′ −1 (p ) (T \ T1) such that

⋆ ′ ′ ′ πX (KX +∆)+ D = KX +∆ . (3.1.13) Since T is a torus, by applying [CP17], we have ′ κ(KX′ +∆ ) > κ(KF +∆F ).

0 ′ ′ Let m ∈ N be a sufficiently divisible number and let s ∈ H (X , mKX′/T + m∆ ). Thanks to ′ ′ −1 (3.1.13) and the fact that D is supported in (p ) (T \ T1), s induces an element 0 ⋆ sT ∈ H (T1,τY (p⋆(mKX/Y + m∆))). 2 Since (p (mK + m∆), h) is hermitian flat on Y , ks k ⋆ (t) is a psh function on t ∈ T . ⋆ X/Y 1 T (πY ) h 1 As codim (T \ T ) > 2, ks k ⋆ (t) is thus constant with respect to t ∈ T . Let r be the T 1 T πY h 1 degree of the cover τY . Since τY is a non-ramified cover on T1, sT induces an element s ∈ 0 −1 H (p (Y \ (∪C )),mrK + mr∆). As ks k ⋆ (t) is constant, by using the same argument 1 i X/Y T τY h 0 e as in the subcase 3.0.1, s can be extended to as an element in H (X,mrKX/Y + mr∆). (3.1.1) is thus proved by using the same argument as in the end of Subcase 3.0.1. e

Our next job is to establish the claim used in the proof of our main result, which is a consequence of the volume estimate inequality (or the holomorphic Morse inequalities).

10 ′ ′ Proof of the claim. Thanks to [PT14], we know that det p⋆(mKX′/Y ′ + m∆ ) is pseudo-effective ′ ′ ′ on Y . Let A be the nef part of the Zariski decomposition of det p⋆(mKX′/Y ′ + m∆ ). Set B := ⋆ (πY ◦ π) OP1 (1). As B is semiample, we have ′ ′ 0 6 A · B 6 c1(det p⋆(mKX′/Y ′ + m∆ )) · c1(B)

′ ′ ⋆ = (πY )⋆(c1(det p⋆(mKX′/Y ′ + m∆ ))) · π c1(OP1 (1))

⋆ = c1(det p⋆(mKX/Y + m∆)) · π c1(OP1 (1)) = 0, where the last equality a consequence of (3.1.4). Then we have A · B = 0. (3.1.14)

′ L·A + + Let L be an ample line bundle on Y and set c := 2L·B ∈ Q . For any τ ∈ Q small enough, thanks to (3.1.14) and the choice of c, the basic volume estimate (cf. for example [Dem12, 8.4] or [Laz04, Thm 2.2.15]) implies that vol(A + τL − cB) > (A + τL)2 − 2c(A + τL) · B

> 2τ(L · A − cL · B)+ o(τ) > 0. Therefore A + τL − cB is big for any τ ∈ Q+. Letting τ → 0+, A − cB is pseudoeffective. Then ′ ′ det p⋆(mKX′/Y ′ + m∆ ) − cB is pseudo-effective and the claim is proved.

4. Appendix

In this appendix, we will gather two standard results which should be well-known to experts.

4.1. Proposition.[Kaw82a, CH11] Let p : X → Y be a fibration from a n-dimensional projective manifold to a K3 surface, and let ∆ be an effective klt Q-divisor on X. Assume that Theorem 1.1 holds for dim X 6 n − 1. If κ(KX + ∆) > 1, then

κ(KX + ∆) > κ(KF +∆F ), where F is the generic fiber of p and ∆F =∆|F .

Proof. We use here the argument in [CP17, Prop 3.7]. Modulo desingularization, we can assume that the Iitaka fibration of KX +∆ is a morphism between two projective manifolds ϕ : X → W .

ϕ / X ❆ W ❆❆ ❆❆ p ❆❆ ❆ Y

Let G be the generic fiber of ϕ and set ∆G := ∆|G. Then

κ(KG +∆G) = 0. (4.1.1) Let p : G → p(G) be the restriction of p on G. We will analyze next among three cases which may occur.

Case 1: We assume that p(G) projects onto Y ; then we argue as follows. Let p : G → Y be e e 11 the Stein factorization of p : G → Y :

p / G ❄ ? Y ❄❄ ⑧⑧ ❄❄ ⑧⑧ e ❄❄ ⑧⑧s p ❄ ⑧⑧ Y

After desingularization p, we can assume that Ye is smooth. Let Gt be the generic fiber of p. By assumption, Theorem1.1 holds for G → Y . Therefore (4.1.1) implies that e e e

κ(Ke Gt +∆Gt ) = 0. (4.1.2) We estimate next the dimension of G. Let F be the generic fiber of p : X → Y . By restricting ϕ on F , we obtain a morphism

ϕt : F → V where V is a subvariety of W . Let V → V be the Stein factorization of ϕt. e ϕt / F ❄ ? V ❄❄ ⑧⑧ ❄❄ ⑧⑧ e ❄❄ ⑧⑧ ϕt ❄ ⑧⑧ V

Since G is generic, we infer that the generic fibere of p coincides with the generic fiber of ϕt. Combining this with (4.1.2), then [Uen75, Thm 5.11] implies that e e κ(KF +∆F ) 6 dim V = dim F − dim Gt. Therefore we have e

dim Gt 6 dim F − κ(KF +∆F ) and thus we infer that

dim G = dim Gt + dim Y 6 dim F − κ(KF +∆F )+dim Y = dim X − κ(KF +∆F ).

Finally, by construction ofe the Iitaka fibration, dim G = dim X − κ(KX + ∆); we obtain the inequality

dim X − κ(KX + ∆) 6 dim X − κ(KF +∆F ), and in conclusion κ(KX + ∆) > κ(KF +∆F ). Case 2: We assume that the image p(G) has dimension zero. Since G is connected, p(G) is a point in Y . This means that we can define a map W → Y , which can be assumed to be regular by blowing up W . We have thus the commutative diagram

ϕ / X ❆ W ❆❆ ⑥⑥ ❆❆ ⑥⑥ p ❆❆ ⑥⑥q ❆ ⑥~ ⑥ Y Set t := p(G). Let F be the fiber of p over t. Then F is a generic fiber of p and G is a generic fiber of ϕ : F → ϕ(F ), and by [Uen75, Thm 5.11] we infer that

κ(KF +∆F ) 6 κ(KG +∆G)+dim ϕ(F ) = dim ϕ(F ).

12 Note that ϕ(F ) is the fiber of q over t ∈ Y . We have dim W = dim ϕ(F ) + dim Y . Therefore dim W > κ(KF +∆F )+dim Y . Combining this with the fact that ϕ is the Iitaka fibration, we have thus

κ(KX +∆) = dim W > κ(KF +∆F )+dim Y, and we are done.

Case 3: The remaining case: p(G) is a proper subvariety of Y . ′ ′ Let p(G) be the normalization of p(G). If p(G) is a curve of general type, then κ(KG +∆G) > 1 and we get a contradiction with (4.1.1). If p(G)′ is P1, as G is generic, Y is thus covered by rational curves. We get a contradiction with the assumption that Y is K3. As a consequence, p(G)′ is a torus. Then [p(G)] is a semi-ample class of numerical dimension 1 in Y . Therefore p(G) is a generic fiber of a fibration π : Y → P1. We have thus the following commutative diagram

ϕ / X ❆ W ❆❆ ⑤⑤ ❆❆ ⑤⑤ p◦π ❆❆ ⑤⑤q ❆ ⑤~ ⑤ P1

Set t := p ◦ π(G). Let Xt be the fiber of p ◦ π over t. Then G is the generic fiber of

ϕ|Xt : Xt → ϕ(Xt), and by [Uen75, Thm 5.11] we infer that

κ(KXt +∆Xt ) 6 κ(KG +∆G)+dim ϕ(Xt) = dim ϕ(Xt). 1 Note that ϕ(Xt) is the fiber of q over t ∈ P . We have dim W = dim ϕ(Xt) + 1. Therefore dim W > κ(KXt +∆Xt ) + 1. Combining this with the fact that ϕ is the Iitaka fibration, we have thus

κ(KX +∆X ) = dim W > κ(KXt +∆Xt ) + 1 > κ(KF +∆F ) + 1, where the last inequality comes from the fact that Xt is a fibration over a torus with the generic fiber F . The proposition is thus proved.

4.2. Proposition. Let p : X → Y be a fibration between two projective manifolds. Let F be the generic fiber and let ∆ be a Q-effective klt divisor on X. Set ∆F := ∆|F . If dim Y = 2 and κ(Y ) > 1, then

κ(KX + ∆) > κ(KF +∆F )+ κ(Y ). (4.2.1)

Proof. Since Y is of dimension 2, we can consider its minimal model and assume that Y is smooth with semi-ample canonical bundle.

If κ(Y ) = 2, then KY is big and it is known that (4.2.1) holds.

If κ(Y ) = 1, we can suppose that KY is semi-ample. Then KY induces a fibration π : Y → Z

⋆ to a 1-dimensional variety Z and KY = π A for some ample line bundle A on Z. Let Yz be a generic fiber of π. Then Yz is a 1-torus. Let m ∈ N be a number sufficiently large and let h be the possibly singular hermitian metric on p⋆(mKX/Y + m∆) defined in Theorem2.7. There are two cases.

13 Case 1. iΘdet h(det p⋆(mKX/Y + m∆))|Yz ≡ 0. Thanks to Proposition2.11, the vector bundle

(p⋆(mKX/Y + m∆)|Yz , h) is hermitian flat. Therefore

2 0 |s|h < +∞ for every s ∈ H (Yz,p⋆(mKX/Y + m∆)). (4.2.2) ZYz ⋆ 2 As KY = π A for some ample line bundle on Z, Proposition 2.9 and the L -condition (4.2.2) imply that

κ(X, KX + ∆) > κ(Xz, KX/Y +∆|Xz ) + 1. (4.2.3)

Moreover, by applying [CP17, Kaw82a] to Xz → Yz and the fact that Yz is a torus, we have

κ(Xz, KX/Y +∆|Xz ) > κ(KF +∆F ). (4.2.4) Together with (4.2.3), (4.2.1) is proved.

Case 2. iΘdet h(det p⋆(mKX/Y +m∆))|Yz 0. As Yz is of dimension 1, det p⋆(mKX/Y +m∆)|Yz ′ is ample on Yz. Since KY is semi-ample, we can find some Q-div ∆ > 0 in the same class of ⋆ ′ c · p KY for some c> 0 small enough such that ∆ + ∆ is klt. Then ′ ′ det p⋆(mKX/Y + m∆+ m∆ ) = det p⋆(mKX/Y + m∆) + m∆ is big on Y . By applying for example [Cam04b, CP17], we have ′ κ(KX/Y +∆+∆ ) > κ(KF +∆F ) + 2. ′ ⋆ As c< 1, we know that KX +∆ − (KX/Y +∆+∆ ) = (1 − c) · p KY is Q-effective. Therefore

κ(KX + ∆) > κ(KF +∆F ) + 2, and (4.2.1) is proved.

References

BHPV Barth, Wolf P. and Hulek, Klaus and Peters, Chris A. M. and Van de Ven, Antonius: Compact Complex surfaces Second edition. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 4. Springer-Verlag, Berlin, Ber09 Berndtsson, Bo: Curvature of vector bundles associated to holomorphic fibrations. Ann. of Math. (2) 169 (2009), no. 2, 531-560. BP08 Berndtsson, Bo and P˘aun, Mihai: Bergman kernels and the pseudoeffectivity of relative canonical bundles. Duke Math. J., 145(2):341-378, 2008. BP10 Berndtsson, Bo and P˘aun, Mihai: Bergman kernels and subadjunction. arxiv: 1002.4145v1

Bir09 Birkar, Caucher: The Iitaka conjecture Cn,m in dimension six. Compos. Math. 145 (2009), no. 6, 1442-1446. BC15 Birkar, Caucher and Chen, Jungkai Alfred: Varieties fibred over abelian varieties with fibres of log general type. Adv. Math. 270 (2015), 206-222. Cam04a Campana, Fr´ed´eric: Orbifoldes premire classe de Chern nulle (French. English summary) [Orb- ifolds of zero first Chern class] The Fano Conference, 339-351, Univ. Torino, Turin, 2004. Cam04b Campana, Fr´ed´eric: Orbifolds, special varieties and classification theory. Ann. Inst. Fourier (Grenoble) 54 (2004), no. 3, 499-630. CC14 Campana, Fr´ed´eric and Benoˆıt, Claudon: Abelianity conjecture for special compact Khler 3-folds. (English summary) Proc. Edinb. Math. Soc. (2) 57 (2014), no. 1, 55-78. CP17 Cao, Junyan and P˘aun, Mihai: Kodaira dimension of algebraic fiber spaces over Abelian varieties Invent. Math. 207 (2017), no. 1, pp 345–387

14 CH11 Chen, Jungkai Alfred and Hacon, Christopher D.: Kodaira dimension of irregular varieties Invent. Math. 186, (2011), 481–500 Dem12 Demailly, Jean-Pierre Analytic methods in algebraic geometry. Surveys of Modern Mathematics, 1. International Press, Somerville, MA; Higher Education Press, Beijing, 2012. viii+231 pp. Fuj16 Fujino, Osamu: Direct images of relative pluricanonical bundles Algebr. Geom. 3 (2016), no. 1, 50-62. Fuj78 Fujita, Takao: On K¨ahler fiber spaces over curves. J. Math. Soc. Japan 30, 779–794 (1978) Fuj79 Fujita, Takao: On Zariski problem Proc. Japan Acad. Ser. A Math. Sci. 55 (1979), no. 3, 106110. GKP15 Kebekus, Stefan; Greb, Daniel and Peternell, Thomas: Singular spaces with trivial canonical class To appear in: Minimal models and extremal rays proceedings of the conference in honor of Shigefumi Moris 60th birthday, Advanced Studies in Pure Mathematics, Kinokuniya Publishing House, Tokyo. Hor10 H¨oring, Andreas: Positivity of direct image sheavesa geometric point of view Enseign. Math. (2) 56 (2010), no. 1-2, 87-142. KP17 Kov´acs, S´andor J. and Patakfalvi, Zsolt: Projectivity of the moduli space of stable log-varieties and subadditivity of log-Kodaira dimension. J. Amer. Math. Soc. 30 (2017), no. 4, 959–1021. Gri70 Griffiths, Phillip A.: Periods of integrals on algebraic manifolds III. Publ. Math. IHES, 38, 125– 180 (1970) HPS16 Hacon, Christopher; Popa, Mihnea and Schnell, Christian: Algebraic fiber spaces over abelian varieties: around a recent theorem by Cao and Paun arXiv:1611.08768 Kaw82a Kawamata, Yujiro: Kodaira dimension of algebraic fiber spaces over curves Invent. Math. 66 (1982), no. 1, 57-71. Kaw82b Kawamata, Yujiro Characterization of Abelian varieties Compositio Mathematica 43 (1981), no. 2, 253-276 Kol85 Koll´ar, J´anos Subadditivity of the Kodaira dimension: fibers of general type Algebraic geometry, Sendai, 1985, 361-398, Adv. Stud. Pure Math., 10, North-Holland, Amsterdam, 1987. KM98 Koll´ar, J´anos and Mori, Shigefumi: Birational geometry of algebraic varieties With the collabo- ration of C. H. Clemens and A. Corti. Translated from the 1998 Japanese original. Cambridge Tracts in Mathematics, 134. Cambridge University Press, Cambridge, 1998. viii+254 pp. Laz04 Lazarsfeld, Robert Positivity in algebraic geometry. I. Classical setting: line bundles and linear series. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], 48. Springer-Verlag, Berlin, 2004. PT14 P˘aun, Mihai and Takayama, Shigeharu: Positivity of twisted relative pluricanonical bundles and their direct images arxiv 1409.5504 Pau16 P˘aun, Mihai: Singular Hermitian metrics and positivity of direct images of pluricanonical bundles arXiv:1606.00174 Rau15 Raufi, Hossein: Singular hermitian metrics on holomorphic vector bundles, Arkiv f¨or Matematik, October 2015, Volume 53, Issue 2, pp 359–382 Uen75 Ueno, Kenji: Classification theory of algebraic varieties and compact complex spaces, Notes written in collaboration with P. Cherenack. Lecture Notes in Mathematics, Vol. 439. Springer-Verlag, Berlin-New York, 1975. xix+278 pp. Vie83 Viehweg, Eckart: Weak positivity and the additivity of the Kodaira dimension for certain fibre spaces, Algebraic varieties and analytic varieties (Tokyo, 1981), 329-353, Adv. Stud. Pure Math., 1, North-Holland, Amsterdam, 1983. Vie95 Viehweg, Eckart: Quasi-projective moduli for polarized manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 30. Springer-Verlag, Berlin, 1995. viii+320 pp.

15 Junyan Cao [email protected] Universit´eParis 6 , Institut de Math´ematiques de Jussieu, 4, Place Jussieu, Paris 75252, France

16